Maxima and Minima of Functions: Finding Absolute and Local Extrema - Prof. Jon W. Lamb | Study notes Mathematics

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APPLICATIONS OF DIFFERENTIATION

4.2 MAXIMUM AND MINIMUM VALUES

I. Remember that the maximum or minimum of a function is the y-value!

A. Plurals: maxima or minima

B. Extrema: all maximum and minimum values

II. The absolute (global) maximum of a function is the (one) largest value of the function on a given

interval

A. May occur at more than one point in the interval: y = sin x has an absolute maximum of 1; it

occurs at x = 2k , k

ππ

+∈]

B. May occur at an endpoint: has an absolute maximum of 16; it occurs at

yxon[4,3]=−

x = -4. This is not a local maximum; see IV below.

C. Some functions do not have an absolute maximum:

ye=

III. The absolute (global) minimum of a function is the smallest value of the function on a given interval

A. May occur at more than one point in the interval: has an absolute minimum of

yx 16x=−

-64 at

x8=±

B. May occur at an endpoint: has absolute minimum of 0; it occurs at x = 0

yx=

C. Some functions do not have an absolute minimum: y = ln x

IV. The local (relative) maximum of a function is the largest value of the function in a small

neighborhood (an open interval) containing the maximum point

A. May occur at more than one point in the interval: y = cos x has a local maximum of 1; it

occurs at x = x2k,k

=∈]

B. Cannot occur at an endpoint because the function does not exist in an open interval containing

the point has no local maxima even though it does have an absolute

yxon[4,3]=−

maximum

C. Some functions do not have a local maximum:

yx=

V. The local (relative) minimum of a function is the smallest value of the function in a small neighbor-

hood (an open interval) containing the minimum point: y = | x | has a local minimum of 0 at x = 0

A. May occur at more than one point in the interval: has a local minimum

43 2

y.25x x 5x=+−

of -93.75 at x = -5 and a local minimum of x = -8 at x = 2

B. Cannot occur at an endpoint because the function does not exist in an open interval containing

the point: has no local minima, even though it does have an absolute maximum

yx=

C. Some functions do not have a local minimum: has no extrema of any kind

yx=

VI. Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute

maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [ a, b].

A. An absolute extremum is not a local extremum if it occurs at an endpoint: yx=

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APPLICATIONS OF DIFFERENTIATION

4.2 MAXIMUM AND MINIMUM VALUES

I. Remember that the maximum or minimum of a function is the y-value!

A. Plurals: maxima or minima

B. Extrema: all maximum and minimum values

II. The absolute (global) maximum of a function is the (one) largest value of the function on a given

interval

A. May occur at more than one point in the interval: y = sin x has an absolute maximum of 1; it

occurs at x = 2k , k

+ π ∈ ]

B. May occur at an endpoint: has an absolute maximum of 16; it occurs at

2 y = x on [ −4, 3]

x = -4. This is not a local maximum; see IV below.

C. Some functions do not have an absolute maximum:

x y =e

III. The absolute (global) minimum of a function is the smallest value of the function on a given interval

A. May occur at more than one point in the interval: has an absolute minimum of

4 2 y = x −16x

-64 at x = ± 8

B. May occur at an endpoint: y = x has absolute minimum of 0; it occurs at x = 0

C. Some functions do not have an absolute minimum: y = ln x

IV. The local (relative) maximum of a function is the largest value of the function in a small

neighborhood (an open interval) containing the maximum point

A. May occur at more than one point in the interval: y = cos x has a local maximum of 1; it

occurs at x = x = 2k π, k∈ ]

B. Cannot occur at an endpoint because the function does not exist in an open interval containing

the point has no local maxima even though it does have an absolute

2 y = x on [ −4, 3]

maximum

C. Some functions do not have a local maximum:

3 y =x

V. The local (relative) minimum of a function is the smallest value of the function in a small neighbor-

hood (an open interval) containing the minimum point: y = | x | has a local minimum of 0 at x = 0

A. May occur at more than one point in the interval: has a local minimum

4 3 2 y = .25x + x −5x

of -93.75 at x = -5 and a local minimum of x = -8 at x = 2

B. Cannot occur at an endpoint because the function does not exist in an open interval containing

the point: y = x has no local minima, even though it does have an absolute maximum

C. Some functions do not have a local minimum: has no extrema of any kind

3 y =x

VI. Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute

maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [ a, b].

A. An absolute extremum is not a local extremum if it occurs at an endpoint: y = x

VII. Fermat’s Theorem: If f has a local maximum or minimum at c, and if f’(c) exists, then f’(c) = 0.

The converse of Fermat’s Theorem is not necessarily true!!

A. If , but f has no minimum or maximum

3 f(x) = x , then f '(0) = 0

B. If f(x) =| x |, then f(0) = 0, but f’(0) does not exist

VIII. A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c)

does not exist. [This is why I have been insisting on factored form for derivatives!]

A. Another way to state Fermat’s Theorem: If f has a local maximum or minimum at c, then c is a

critical number of f. NOT REVERSIBLE !!

B. Find all critical numbers of. This was #5 on the Differentiation Practice worksheet.

x

y

4x 1

We found in factored form!

3 2

dy 2x(2x 1)

dx (4x 1)

dy (^3) 0 2x(2x 1) 0 dx

So, either.

3 x = 0 or 2x − 1 = 0.

2x 1 0 x x. 2 2

Check to make certain the denominator is not equal to 0 for either of these values.

is undefined if. Hmmmmm...a critical

4x 1 0 x x. 4 4

number must be in the domain of the function, so that our work in the last step was

unnecessary! The critical numbers are x = 0 and 3

x. 2

C. Find all critical numbers of y = x ln(x): This was #23 on the DP worksheet.

dy 1 1 ln x 1 0 ln x 1 e x x. dx e

− = + = ⇒ = − ⇒ = ⇒ = ≈

D. Find all critical numbers of. This was #18 on the DP worksheet.

2 cos x y 1 sin x

however some of these values will make sin x = 1 and

dy cos x 0 x k , k , dx 2

= = ⇒ = + π ∈ ]

the original function will be undefined. sin x 1 x 2k , k. The critical numbers

= ⇒ = + π ∈ ]

are

x 2k , k. 2

= + π ∈ ]

E. Find all critical numbers of y =(tan x)(cot x). This was #11 on the DP worksheet.

Since all numbers which satisfy the original function are critical numbers. The function

dy 0, dx

is undefined whenever cos x = 0 or sin x = 0, so we must rule out all multiples of. The critical

numbers are all real numbers except

k x , k. 2

= ∈ ]

F. Find all critical numbers of. This was #29 on the DP worksheet. We found

1 x x

y e e

, but we need to put the derivative in fractional form in order to find the

1 x

2 x

dy e 1

dx (^) x e

critical numbers.

1 1 x x (^1 ) x x

x 2 x 2 2 x 2 x 2 x

dy e 1 e e x 0 e e x 0 x e e. dx (^) x e x e

Maxima and Minima of Functions: Finding Absolute and Local Extrema - Prof. Jon W. Lamb, Study notes of Mathematics

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Download Maxima and Minima of Functions: Finding Absolute and Local Extrema - Prof. Jon W. Lamb and more Study notes Mathematics in PDF only on Docsity!

APPLICATIONS OF DIFFERENTIATION

4.2 MAXIMUM AND MINIMUM VALUES

I. Remember that the maximum or minimum of a function is the y-value!

A. Plurals: maxima or minima

B. Extrema: all maximum and minimum values

II. The absolute (global) maximum of a function is the (one) largest value of the function on a given

interval

A. May occur at more than one point in the interval: y = sin x has an absolute maximum of 1; it

occurs at x = 2k , k

+ π ∈ ]

B. May occur at an endpoint: has an absolute maximum of 16; it occurs at

x = -4. This is not a local maximum; see IV below.

C. Some functions do not have an absolute maximum:

III. The absolute (global) minimum of a function is the smallest value of the function on a given interval

A. May occur at more than one point in the interval: has an absolute minimum of

-64 at x = ± 8

B. May occur at an endpoint: y = x has absolute minimum of 0; it occurs at x = 0

C. Some functions do not have an absolute minimum: y = ln x

IV. The local (relative) maximum of a function is the largest value of the function in a small

neighborhood (an open interval) containing the maximum point

A. May occur at more than one point in the interval: y = cos x has a local maximum of 1; it

occurs at x = x = 2k π, k∈ ]

B. Cannot occur at an endpoint because the function does not exist in an open interval containing

the point has no local maxima even though it does have an absolute

maximum

C. Some functions do not have a local maximum:

V. The local (relative) minimum of a function is the smallest value of the function in a small neighbor-

hood (an open interval) containing the minimum point: y = | x | has a local minimum of 0 at x = 0

A. May occur at more than one point in the interval: has a local minimum

of -93.75 at x = -5 and a local minimum of x = -8 at x = 2

B. Cannot occur at an endpoint because the function does not exist in an open interval containing

the point: y = x has no local minima, even though it does have an absolute maximum

C. Some functions do not have a local minimum: has no extrema of any kind

VI. Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute

maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [ a, b].

A. An absolute extremum is not a local extremum if it occurs at an endpoint: y = x

VII. Fermat’s Theorem: If f has a local maximum or minimum at c, and if f’(c) exists, then f’(c) = 0.

The converse of Fermat’s Theorem is not necessarily true!!

A. If , but f has no minimum or maximum

B. If f(x) =| x |, then f(0) = 0, but f’(0) does not exist

VIII. A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c)

does not exist. [This is why I have been insisting on factored form for derivatives!]

A. Another way to state Fermat’s Theorem: If f has a local maximum or minimum at c, then c is a

critical number of f. NOT REVERSIBLE !!

B. Find all critical numbers of. This was #5 on the Differentiation Practice worksheet.

x

y

4x 1

We found in factored form!

dy 2x(2x 1)

dx (4x 1)

So, either.

Check to make certain the denominator is not equal to 0 for either of these values.

is undefined if. Hmmmmm...a critical

number must be in the domain of the function, so that our work in the last step was

unnecessary! The critical numbers are x = 0 and 3

C. Find all critical numbers of y = x ln(x): This was #23 on the DP worksheet.

D. Find all critical numbers of. This was #18 on the DP worksheet.

however some of these values will make sin x = 1 and

= = ⇒ = + π ∈ ]

the original function will be undefined. sin x 1 x 2k , k. The critical numbers

= ⇒ = + π ∈ ]

are

= + π ∈ ]

E. Find all critical numbers of y =(tan x)(cot x). This was #11 on the DP worksheet.

Since all numbers which satisfy the original function are critical numbers. The function

is undefined whenever cos x = 0 or sin x = 0, so we must rule out all multiples of. The critical

numbers are all real numbers except

= ∈ ]

F. Find all critical numbers of. This was #29 on the DP worksheet. We found

, but we need to put the derivative in fractional form in order to find the

critical numbers.